A positive integer has the property that m2 is expressible in the form 4n2 – 5n + 16 where is an integer (of any sign). Find the maximum possible value of |m – n|.
Hint 1: Start by analyzing the initial conditions and setting up the basic equations. m2 = 4n2 – 5n + 16.
Hint 2: Look for algebraic properties, symmetry, or geometric theorems to simplify. => 4n2 – 5n + 16 – m2 = 0.
Hint 3: Proceed with the final algebraic steps to solve the system. => 8n solve for the final value 16m 2 – 231.
Step 1: m2 = 4n2 – 5n + 16
Step 2: => 4n2 – 5n + 16 – m2 = 0
Step 3: => 8n = 5 16m 2 – 231
Step 4: Let D2 = 16m2 – 231 and D > 0
Step 5: => (4m + D) (4m – D) = 231 = 7 × 11 × 13
Step 6: 4m + D = 231 and 4m – D = 1
Step 7: then = 29 and D = 115, = 15
Step 8: => |m – n| = 14
Step 9: 4m + D = 77 and 4m – D = 3
Step 10: => = 10, D = 37 and = –4
Step 11: => |m – n| = 14
Step 12: 4m + D = 33 and 4m – D = 7
Step 13: => = 5, D = 13 and = –1
Step 14: => |m – n| = 6
Step 15: 4m + D = 21 and 4m – D = 11
Step 16: => = 4, D = 5 and = 0
Step 17: Hence |m – n| = 4
Step 18: Maximum value of |m – n| = 14
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